Quantitative Reasoning

Activity – Analyzing the Results

• Why aren’t all the results the same?

• How do we compare results?

• What kind of errors occurred?

• Is our error small or big?

• Is our result precise, accurate or what?

• So, do our results agree?

Why aren’t all the results the same?

• Useful questions to ask, if results don’t agree:

– Which object did you measure?

– What units where you using?

– What is your estimated error?

How do we compare results?

• Don’t compare apples with oranges!

• Need to use the same units

• 1 inch = 2.54 cm

• To get inches from centimeters, divide by 2.54

• To get centimeters from inches, multiply by 2.54

What kind of errors occurred?• There are systematic and random errors

• To beat down random errors, measure the same thing many times, and the errors will even out, i.e. the overall error will be smaller

• The systematic error can be reduced by doing a better experiment, or understanding your instruments better (miscalibrations etc.)

• Human error is not an acceptable error source in science! It just means you are a bad experimenter.

Is our error small or big?• It depends!

• If you have a small error and the measured length is also small, you might have a huge error!

• Use percentages: – Percent error = (estimated error)/(result) x 100%– Example: 51.3 cm ± 0.2 cm gives

– Percent error = (0.2 cm)/(51.3cm) x 100 % = 0.4 % (This is a pretty small error)

Is our result precise or accurate or what?• Two different concepts: precision and accuracy!

• High precision means small error

• High accuracy means close to an accepted value

• Examples: * * * * high precision, high accuracy

* * * * high precision, low accuracy

* * * * low precision, high accuracy

* * * * low precision, low accuracy

accepted value

So, do our results agree?

• Results agree, if they are within the error margins of each other

• Examples:

| O | | O |

values very different, but errors large: agreement!

| O | | O |

values closer, but errors smaller: no agreement!

Quantitative Reasoning

• Amazingly powerful tool to understand the world around us

• Fundamentals:– Ratios– Graphs– Area &Volume– Scaling– Arithmetical statements

Achieving Scientific Literacy(Arons Article)

• Two types of knowledge– Declarative (Learned Facts, “book knowledge”)– Operative (actually knowing how to solve

problems)

• Trouble with GenEd courses– Too much in too little time– Getting a “feeling” for the subject doesn’t work– Need to understand the underpinnings first (area,

volume, scaling, energy, atoms,…)

Scaling

• Often one is interested in how quantities change when an object or a system is enlarged or shortened

• Different quantities will change by different factors!

• Typical example: how does the circumference, surface, volume of a sphere change when its radius changes?

How does it scale?

• Properties of objects scale like the perimeter, the area or the volume– Mass scales like the volume (“more of the same

stuff”)– A roof will collect rain water proportional to its

surface area